Towards a Characterization of Random Serial Dictatorship

TL;DR

This paper investigates the characterization of Random Serial Dictatorship (RSD) for any number of agents, extending known results for small cases and exploring the limitations of current proof techniques.

Contribution

It extends the characterization of RSD to arbitrary numbers of agents by analyzing the underlying matrix rank and providing counterexamples to previous proof approaches.

Findings

RSD is characterized by axioms for up to 3 agents.

Extending this characterization to more agents involves complex matrix rank conditions.

Counterexamples show certain proof methods are insufficient for generalization.

Abstract

Random serial dictatorship (RSD) is a randomized assignment rule that - given a set of $n$ agents with strict preferences over $n$ houses - satisfies equal treatment of equals, ex post efficiency, and strategyproofness. For $n \le 3$, Bogomolnaia and Moulin (2001) have shown that RSD is characterized by these axioms. Extending this characterization to arbitrary $n$ is a long-standing open problem. By weakening ex post efficiency and strategyproofness, we reduce the question of whether RSD is characterized by these axioms for fixed $n$ to determining whether a matrix has rank $n^2 n!^n$. We provide computer-generated counterexamples to show that two other approaches for proving the characterization (using deterministic extreme points or restricted domains of preferences) are inadequate.